On formulaes of integration by parts for special type of power functions

O. M. Buhrii

doi:10.15330/ms.45.2.118-131

We investigate properties of the function

$\psi_{\alpha}(u)$ , where

$\alpha>0$ ,

$u\in C^{m}(\overline{G})$ or

$u\in W^{m,p}(G)$ ,

$G\subset R^N$ ,

$\psi_{\alpha}$ is some power function, for example,

$\psi_{\alpha}(s)=|s|^{\alpha-1}s$ . In particular, some formulaes of integration by parts and differentiation for observed functions are proved.

1. Metafune G., Spina C. An integration by part formula in Sobolev spaces// Mediterr. J. Math. – 2008. – V.5. – P. 357–369.

2. Lions J.-L., Strauss W.A. Some non-linear evolution equations// Bulletin de la S. M. F. – 1965. – V.93. – P. 43–96.

3. Gajewski H., Groger K., Zacharias K. Nichtlineare operatorgleichungen und operatordifferentialgleichungen. – Moscow, 1978, 336 p. (in Russian)

4. Ladyzenskaya O.A., Uraltseva N.N. Linear and quasilinear elliptic equations. – Moscow, 1973, 576 p. (in Russian)

5. Kinderlehrer D., Stampacchia G. An introduction to variational inequalities and their applications – Moscow, 1983, 256 p. (in Russian)

6. Baiocchi C., Capelo A. Variational and quasivariational inequalities. Applications to free boundary problems. – Moscow, 1988, 448 p. (in Russian)

7. Kolmogorov A.N., Fomin S.V. Theory of functions and functional analysis. – Moscow, 1972, 496 p. (in Russian)

8. Bystrom J. Sharp constants for some inequalities connected to the p-Laplace operator// Jour. of Ineq. in Pure and Appl. Math. – 2005. – V.6, Issue2. – Article 56.

9. Gorodetskyi V.V., Nagnybida N.I., Nastasiev P.P. Methods of solving of function analysis problems. – Kyiv, 1990, 479 p. (in Russian)

	On formulaes of integration by parts for special type of power functions (in Ukrainian)
Author	O. M. Buhrii ol_buhrii@i.ua Ivan Franko National University of Lviv
Abstract	We investigate properties of the function $\psi_{\alpha}(u)$ , where $\alpha>0$ , $u\in C^{m}(\overline{G})$ or $u\in W^{m,p}(G)$ , $G\subset R^N$ , $\psi_{\alpha}$ is some power function, for example, $\psi_{\alpha}(s)=\|s\|^{\alpha-1}s$ . In particular, some formulaes of integration by parts and differentiation for observed functions are proved.
Keywords	integration by parts formulae; differentiation formulae; exponential functions
DOI	doi:10.15330/ms.45.2.118-131
Reference	1. Metafune G., Spina C. An integration by part formula in Sobolev spaces// Mediterr. J. Math. – 2008. – V.5. – P. 357–369. 2. Lions J.-L., Strauss W.A. Some non-linear evolution equations// Bulletin de la S. M. F. – 1965. – V.93. – P. 43–96. 3. Gajewski H., Groger K., Zacharias K. Nichtlineare operatorgleichungen und operatordifferentialgleichungen. – Moscow, 1978, 336 p. (in Russian) 4. Ladyzenskaya O.A., Uraltseva N.N. Linear and quasilinear elliptic equations. – Moscow, 1973, 576 p. (in Russian) 5. Kinderlehrer D., Stampacchia G. An introduction to variational inequalities and their applications – Moscow, 1983, 256 p. (in Russian) 6. Baiocchi C., Capelo A. Variational and quasivariational inequalities. Applications to free boundary problems. – Moscow, 1988, 448 p. (in Russian) 7. Kolmogorov A.N., Fomin S.V. Theory of functions and functional analysis. – Moscow, 1972, 496 p. (in Russian) 8. Bystrom J. Sharp constants for some inequalities connected to the p-Laplace operator// Jour. of Ineq. in Pure and Appl. Math. – 2005. – V.6, Issue2. – Article 56. 9. Gorodetskyi V.V., Nagnybida N.I., Nastasiev P.P. Methods of solving of function analysis problems. – Kyiv, 1990, 479 p. (in Russian)
Pages	118-131
Volume	45
Issue	2
Year	2016
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

On formulaes of integration by parts for special type of power functions (in Ukrainian)